Information• A (finite) set states S, probabilities πs> 0for each s ∈ S, and dates t = 0, 1, . . . , T.• At each date t a collection of subsets of S,Ft= {A1t, A2t, · · · Aktt}, such tha t AitTAjt= ∅if j 6= i andSiAit= S. (Partition)– F0= {S}.– Each A ∈ FTcontains exactly one state.– Ait, i = 1, . . . , kt, are the events at t.• Information structure: F = {F0, F1, · · · FT}.• Total recall: If A ∈ Ft, and t0< t thereexists an A0∈ Ft0such that A ⊂ A0.• Trees1• Stochastic process: A collection of randomvariables yt(s) for t = 0, · · · T.• Stochastic process is adapted to F if foreach A ∈ Ft, yt(s) = yt(s0) for each s ∈ Aand s0∈ A. yt(A) ≡ yt(s), s ∈ A.• y ≥ 0 (positive) if yt(s) ≥ 0 for each (t, s).y > 0 (positive and non-zero) if y ≥ 0 andy 6= 0. y >> 0 (strictly positive) if yt(s) > 0for each (t, s).• For each j = 1, . . . J, a security is an adapteddividend process xjt, t = 1, . . . T. The divi-dend paid at t is received by the agent thatheld the security from t − 1 to t.• The (ex dividend) price of this security isan adapted process pjt.2Strategies• A strategy consist of J adapted processesh1, · · · , hJ. hjtdenotes the amount held fromt to t + 1 of security j.• H the set of all strategies.• The dividend of the strategy is the processzht= (ht−1− ht) · pt+ ht−1· xt≡PJj=1[pjt(hjt−1− hjt) + hjt−1xjt], for t ≥ 1.• zhis adapted.• The cost of strategy h iszh0= p0· h0≡Xjpj0hj0.3Marketed subspace• Mp= {y : y = zhfor some h ∈ H}.• Mpis a linear space.– Complete markets if any adapted y ∈Mp.– Incomplete markets.4Dynamic hedging• No dividends paid until period 2.• Prices of the two assets in parenthesis in periods 0,1. Dividends in parenthesisin period 2.• Hedge: z0= z1= 0, z2 = (0,1,0,0)5Arbitrage• The law of one price holds ifzh= zh0⇒ p0· h0= p0· h00.• Law of one price ⇔ every portfolio strategywith zero payoff has zero price.• If the law of one price holds we may definea linear functional q : Mp→ R defined by:q(z) = p0· h0for any strategy h such thatzh= z. q is the payoff pricng functional.• A strong arbitrage is a strategy h withp0· h0< 0 and zh≥ 0.• An arbitrage is a strategy h that is either astrong arbitrage or satisfies p0· h0= 0 andzh> 0.6• The payoff pricing functional is strictly pos-itive (q(z) > 0 for every z > 0, z ∈ M) ⇔there is no arbitrage.• The payoff pricing functional is positive (q(z) ≥0 for every z ≥ 0, z ∈ M) ⇔ there is nostrong arbitrage.• A one-period strong arbitrage at a n eventAitat t < T is a portfolio h such that, foreach s ∈ Ait,[pt+1(s) + xt+1(s)] · h ≥ 0,and pt(s) · h < 0.• A one-period arbitrage at an event Aitatt < T is a portfolio h that is either a one-period strong arbitrage or that pt(s) · h = 0and [pt+1(s) + xt+1(s)] · h > 0.• No one-period arbitrage ⇔ no arbitrage.7Complete markets• The immediate successors of an event A ∈Ftare all the events B ∈ Ft+1such thatB ⊂ A.• ι(A) ≡ the number of immediate successorsof an event A.• The one-period payoff matrix in event A ∈Ft, t < T, is the matrix with entriespjt+1(B) + xjt+1(B), for j = 1, . . . J and Ban immediate successor of A.• Markets are complete ⇔ for all events A ∈Ft, t < T the one period payoff matrix hasrank ι(A).• J ≥ ι(Ait) whenever t < T.8Event prices• Given an event At∈ Ft, t > 1, consider a se-curity that has a dividend process yAtwithyAtτ(s) = 0 if τ 6= t, yAtt(s) = 0 if s 6∈ At, andyAtt(s) = 1 if s ∈ At.• If complete markets and the law of oneprice prevails then q(yAt) is called the priceof an elementary (Arrow-Debreu) claim as-sociated with At, or the price of the eventAt.• q(yA0) = q(yS) = 1.9• For any security xj, we may write:xj=XtXAt∈Ftxjt(At)yAt.•q(xj) =TXt=1XAt∈Ftxjt(At)q(yAt). (1)• For any strategy hp0· h0=TXt=1XAt∈Ftzht(At)q(yAt).• Suppose you follow strategy of buying anasset at t < T and selling at t + 1. Then:q(yAt)pj(At) =XAt+1⊂At,At+1∈Ft+1q(yAt+1)[pj(At+1) + xj(At+1)].10Implications of no arbitrage• An event price vector is a family qt(At) foreach At∈ Ft, 0 ≤ t ≤ T, with q0(S) = 1.• Event prices q(At) are compatible with (x, p)if for every j = 1, . . . , J and every (t, At),0 ≤ t ≤ T and At∈ Ftqt(At)pj(At) =Xt<τ≤TXAτ⊂At,Aτ∈Fτqt(Aτ)xj(Aτ). (2)• Equation (2) holds if and only ifqt(At)pj(At) =XAt+1⊂At,At+1∈Ft+1qt+1(At+1)[xj(At+1) + pj(At+1)].11Two basic results• There exists a strictly positive vectorof event prices consistent with (x, p) ifand only if there is no arbitrage.• If no arbitrage, markets are complete ifand only if there exists a unique posi-tive vector of event prices that are con-sistent with (x, p).12Risk-free return and discount factors• If pjt(s) > 0,rjt+1(s) =pjt+1(s) + xjt+1(s)pjt(s)is the rate of return between t and t + 1 instate s.• rjt+1(s) only depend on the event At+1∈Ft+1that contains s.• An asset j is risk free at (t, s) if rjt+1(s) onlydepends on the event At∈ Ftthat containss.13• If no arbitrage, all risk-free assets at a given(t, s) must have the same ¯rt(s) > 0.• If At∈ Ft, let ¯rt(At) = ¯rt(s), for any s ∈ At.•qt(At) = ¯rt+1(At+1)XAt+1⊂At,At+1∈Ft+1qt+1(At+1)• If a risk-free security exists at each (t, s),the discount factor between zero and t instate s is ρ0(s) = 1 and,ρt(s) = Πtτ=1(¯rτ(s))−1, t ≥ 1– ρt(s) only depends on the event At−1∈Ft−1that contains s.– ρt(s) = ρt+1(s)¯rt+1(s)• If At∈ Ftlet ρt(At) = ρt(s) for any s ∈ At.14An example: model of interest rate• State s = (r0, r1, . . . , rT), where rtis oneplus the one period interest rate prevailingat t.• A one year bond issued at t pays a dividendof 1 at t + 1. The price at t is1rt(s).• A τ year zero coupon bond issued at t paysa dividend of 1 at t + τ .– A τ year zero coupon issued at t is aτ − j bond in period t + j.• Pτt(s) the price of a τ year zero couponbond as of t, (P0t(s)=1). If At∈ Ft,Pτt(At)qt(At) =XAt+1⊂At, At+1∈Ft+1Pτ−1t+1(At+1)qt+1(At+1).15Risk-neutral probabilities• Assume no arbitrage or equivalently thatyou have a set of positive state prices, andthat at each (t, s) a one period risk freesecurity exists.• Recall that {s} ∈ FT, and set q(s) = qT({s}).• Letπ∗(s) =q(s)ρT(s)> 0,• For every A ⊂ S letπ∗(A) =Xs∈Aπ∗(s).16• If AT −1∈ FT −1, thenqT −1(AT −1) = ¯rT(s)Ps∈AT −1q(s)• qT −1(AT